Screw jack system
 

My team is considering building a vertical dual-screw mechanism. We are considering two ways to power it - the first way involves a window motor powering each screw independently. I'm worried that if the motors are not completely identical, minute differences in speed would cause the screws to bind, taking us out of competition.

The other option is to chain the two motors together and power the screws with a horizontal chain. My team's mentor is worried that, because the chain is horizontal, it might run the risk of jumping off the sprockets during competition.

Any advice?

Just make sure you have enough teeth engaged and that you consider an idler or adjustment technique to 1) help you at installation and 2) can take up the slack when changing direction

For something like this you could probably also use timing belts/pulley which would reduce weight as well.

Could you use two guide pins and one screw? You could two pieces of drill rod and use those to guide the shuttle and use one screw two move the shuttle.

GUI I believe this is one of the better ways to accomplish your task. A single timing belt running to both lead screws and the drive motor. More or less the belt would form a triangle with the lead screws at two corners and the motor at the third.
A second alternative, if space allows, is the drive motor with a sprocket, midway between the two screws, driving sprockets attached to the screws. (visualize three ring circus)

Limit switches
 

If you have to use two motors, and you can get an arm from one moving piece to the other, you can do this.

Put two limit switches (micro switches,...) on one piece so that the arm hits one or the other if the two sides are not even (level). Then have the software slow down the motor that is ahead.

Make the arm tip flexible to avoid damage if there is an error

Switches picture
 

Here is a drawing of what I was talkign about in my previous reply (thanks to the admins for enableing PNG attachments so quickly).